Curtis Porter : CRash CouRse in CR Geometry
- Graduate/Faculty Seminar,Uploaded Videos ( 2252 Views )CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartan’s generalization of Klein’s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.
Oliver Tough : The Fleming-Viot Particle System with McKean-Vlasov dynamics
- Probability,Uploaded Videos ( 1332 Views )Quasi-Stationary Distributions (QSDs) describe the long-time behaviour of killed Markov processes. The Fleming-Viot particle system provides a particle representation for the QSD of a Markov process killed upon contact with the boundary of its domain. Whereas previous work has dealt with killed Markov processes, we consider killed McKean-Vlasov processes. We show that the Fleming-Viot particle system with McKean-Vlasov dynamics provides a particle representation for the corresponding QSDs. Joint work with James Nolen.
Eliza O’Reilly : Stochastic and Convex Geometry for Complex Data Analysis
- Colloquium Seminar,Colloquium,Uploaded Videos ( 823 Views )Many modern problems in data science aim to efficiently and accurately extract important features and make predictions from high dimensional and large data sets. Naturally occurring structure in the data underpins the success of many contemporary approaches, but large gaps between theory and practice remain. In this talk, I will present recent progress on two different methods for nonparametric regression that can be viewed as the projection of a lifted formulation of the problem with a simple stochastic or convex geometric description, allowing the projection to encapsulate the data structure. In particular, I will first describe how the theory of stationary random tessellations in stochastic geometry can address the computational and theoretical challenges of random decision forests with non-axis-aligned splits. Second, I will present a new approach to convex regression that returns non-polyhedral convex estimators compatible with semidefinite programming. These works open many directions of future work at the intersection of stochastic and convex geometry, machine learning, and optimization.
Yang Li : On the Donaldson-Scaduto conjecture
- Geometry and Topology ( 708 Views )Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in X \times R^3, where X is an A2-type ALE hyperkähler manifold. We prove this conjecture by solving a real Monge-Ampère equation with singular right hand side. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X \times R^2, where X arises from the Gibbons-Hawking construction. This is joint work in progress with Saman Habibi Esfahani.
Demetre Kazaras : If Ricci is bounded below, then mass is in control!
- Geometry and Topology ( 553 Views )The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I'll use this formula to consider the question: How flat is an asymptotically flat manifold with very little total mass? In the presence of a lower bound on Ricci curvature, we make progress on this question and confirm special cases of conjectures made by Ilmanen and Sormani.
Jerry Yu Fu : A density theorem towards p-adic monodromy
- Number Theory ( 456 Views )We investigate the $p$-adic monodromy of certain kinds of abelian varieties in $\mathcal{A}_{g}$ and prove a formal density theorem for the locus of deformations with big monodromy. Also, we prove that the small monodromy locus of the deformation space of a supersingular elliptic curve is $p$-adic nowhere dense. The approach is based on a congruence condition of $p$-divisible groups and transform of data between the Rapoport-Zink spaces and deformation spaces.
Joe Kileel : Inverse Problems, Imaging, and Tensor Decomposition
- Applied Math and Analysis ( 374 Views )Perspectives from computational algebra and numerical optimization are brought to bear on a scientific application and a data science application. In the first part of the talk, I will discuss cryo-electron microscopy (cryo-EM), an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D projections, recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM presents a particularly rich inverse problem, with unknown orientations, extreme noise, big data and conformational heterogeneity. In particular, this motivates a general framework for statistical estimation under compact group actions, connecting information theory and group invariant theory. In the second part of the talk, I will discuss tensor rank decomposition, a higher-order variant of PCA broadly applicable in data science. A fast algorithm is introduced and analyzed, combining ideas of Sylvester and the power method.
Francis Brown : Periods, Galois theory and particle physics: Amplitudes in high-energy physics
- Gergen Lectures ( 321 Views )In high-energy physics, interactions between fundamental particles can be represented by Feynman graphs. Almost all predictions for particle collider experiments are obtained by computing certain integrals associated to such graphs, called Feynman integrals, and a vast effort in the physics community worldwide is devoted to studying these quantities. Feynman integrals turn out to be periods, and surprisingly many are multiple zeta values. I will survey what is known and not known about these quantities.
Oguz Savk : Bridging the gaps between homology planes and Mazur manifolds.
- Geometry and Topology,Uploaded Videos ( 315 Views )We call a non-trivial homology 3-sphere a Kirby-Ramanujam sphere if it bounds a homology plane, an algebraic complex smooth surface with the same homology groups of the complex plane. In this talk, we present several infinite families of Kirby-Ramanujam spheres bounding Mazur type 4-manifolds, compact contractible smooth 4-manifolds built with only 0-, 1-, and 2-handles. Such an interplay between complex surfaces and 4-manifolds was first observed by Ramanujam and Kirby around nineteen-eighties. This is upcoming joint work with Rodolfo Aguilar Aguilar.
Christine Heitsch : The Combinatorics of RNA Branching
- Mathematical Biology ( 304 Views )Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly, especially for sequences on the scale of viral genomes. However, results from enumerative, probabilistic, analytic, and geometric combinatorics yield insights into RNA structure formation, and suggest new directions in viral capsid assembly.
Jessica Fintzen : Frontiers in Mathematics Lecture 1: Representations of p-adic groups
- Presentations ( 294 Views )The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.
Florian Naef : A real description of brackets and cobrackets in string topology
- Presentations ( 289 Views )Let M be a manifold with non-vanishing vectorfield. The homology of the space of loops in M carries a natural Lie bialgebra structure described by Sullivan as string topology operations. If M is a surface, these operations where originally defined by Goldman and Turaev. We study formal descriptions of these Lie bialgebras. More precisely, for surfaces these Lie bialgebras are formal in the sense that they are isomorphic (after completion) to their algebraic analogues (Schedler's necklace Lie bialgebras) built from the homology of the surface. For higher dimensional manifolds we give a similar description that turns out to depend on the Chern-Simons partition function.
This talk is based on joint work with A. Alekseev, N. Kawazumi, Y. Kuno and T. Willwacher.
David Schwein : Recent progress on the formal degree conjecture
- Number Theory ( 288 Views )The local Langlands correspondence is a dictionary between representations of two kinds of groups: reductive p-adic groups (such as the general linear group) and the absolute Galois groups of p-adic fields. One entry in the dictionary is a conjectural formula of Hiraga, Ichino, and Ikeda for the size of a representation of a p-adic group, its "formal degree", in terms of the corresponding representation of a Galois group. In this talk, after reviewing the broad shape of p-adic representation theory, I'll explain why the conjecture is true for almost all supercuspidals, the fundamental building blocks of the subject.
Curtis Porter : Spinning Black Holes and CR 3-Folds
- Geometry and Topology ( 276 Views )Some physically significant solutions to Einstein's field equations are spacetimes which are foliated by a family of curves called a shear-free null geodesic congruence (SFNGC). Examples include models of gravitational waves that were recently detected, and rotating black holes. The properties of a SFNGC induce a CR structure on the 3-dimensional leaf space of the foliation. The Kerr Theorem says that the family of metrics associated to a SFNGC contains a conformally flat representative iff the corresponding CR structure is embeddable in a real hyperquadric. Using Cartan's method of moving frames, we can classify which Levi-nondegenerate CR 3-folds are embeddable in the hyperquadric.
Ana Barros : Down the Predictability Hole - Searching for Metrics to Understand and Replace Physical Parameterizations of Nonlinear Processes in Atmospheric Models
- Nonlinear and Complex Systems ( 274 Views )Short-term forecast skill in weather forecasting over the last 15 years has been achieved mostly through data assimilation. Predictive ability however has hit barriers that have not been overcome by increasing computer power and model resolution. Model tuning has come out from hiding, and it is arguably ``trending'' in peer-review over the last 2-3 years. The open question is what (and how) to do next. I will address this question relying on two-decades of research on the representation of moist processes in the atmosphere, specifically targeting the following issues: 1) Evaluating Models to Elucidate the Physics that Matter 2) Detecting and Isolating Sources, Sinks and Barriers of Predictability 3) Meeting the Utility Challenge - Projections vs Predictability
Miklos Racz : From trees to seeds: on the inference of the seed from large random trees
- Presentations ( 266 Views )I will discuss the influence of the seed in models of randomly growing trees; in particular, I will focus on the preferential attachment and uniform attachment models. In both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. I will discuss the differences and similarities in proving this for the two models. This is based on joint work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel.
Arthur Szlam : A Total Variation-based Graph Clustering Algorithm for Cheeger Ratio Cuts
- Undergraduate Seminars ( 261 Views )I will discuss a continuous relaxation of the Cheeger cut problem on a weighted graph, and show how the relaxation is actually equivalent to the original problem. Then I will introduce an algorithm which experimentally is very efficient at approximating the solution to this problem on some clustering benchmarks. I will also give a heuristic variant of the algorithm which is faster but often gives just as accurate clustering results. This is joint work with Xavier Bresson, inspired by recent papers of Buhler and Hein, and Goldstein and Osher, and by an older paper of Strang.
Daniel Stern : Spectral shape optimization and new behaviors for free boundary minimal surfaces
- Geometry and Topology ( 259 Views )Though the study of isoperimetric problems for Laplacian eigenvalues dates back to the 19th century, the subject has undergone a renaissance in recent decades, due in part to the discovery of connections with harmonic maps and minimal surfaces. By the combined work of several authors, we now know that unit-area metrics maximizing the first nonzero Laplace eigenvalue exist on any closed surface, and are realized by minimal surfaces in spheres. At the same time, work of Fraser-Schoen, Matthiesen-Petrides and others yields analogous results for the first eigenvalue of the Dirichlet-to-Neumann map on surfaces with boundary, with maximizing metrics induced by free boundary minimal immersions into Euclidean balls. In this talk, I'll describe a series of recent results characterizing the (perhaps surprising) asymptotic behavior of these free boundary minimal immersions (and associated Steklov-maximizing metrics) as the number of boundary components becomes large. (Based on joint work with Mikhail Karpukhin.)
John McCuan : Minimal graphs with jump discontinuities
- Geometry and Topology ( 254 Views )I will discuss some examples of minimal graphs with jump discontinuities in their boundaries. Robert Huff and I constructed these examples in response to a question of John Urbas: Is it possible for a minimal graph over a smooth annular domain to have an isolated jump discontinuity on the inner boundary component? I will also give a brief overview of the boundary consistency problem for Di Giorgi's generalized solutions of the minimal surface equation and discuss this question in that context. The construction of the examples uses the Weierstrass representation and the developing map introduced by Huff in the study of capillary problems.
Evangelia Gazaki : Torsion phenomena for zero-cycles on a product of curves over a number field
- Number Theory ( 252 Views )For a smooth projective variety X over an algebraic number field a conjecture of Bloch and Beilinson predicts that the kernel of the Abel-Jacobi map of X is a torsion group. When X is a curve, this follows by the Mordell-Weil theorem. In higher dimensions however there is hardly any evidence for this conjecture. In this talk I will focus on the case when X is a product of smooth projective curves and construct infinitely many nontrivial examples that satisfy a weaker form of the Bloch-Beilinson conjecture. This relies on a recent joint work with Jonathan Love.
Siqi He : Classification of Nahm Pole Solutions to the KW Equations on $S^1\times\Sigma\times R^+$
- Geometry and Topology ( 251 Views )We will discuss Witten’s gauge theory approach to Jones polynomial by counting solutions to the Kapustin-Witten (KW) equations with singular boundary conditions over 4-manifolds. We will give a classification of solutions to the KW equations over $S^1\times\Sigma\times R^+$. We prove that all solutions to the KW equations over $S^1\times\Sigma\times R^+$ are $S^1$ direction invariant and we give a classification of the KW monopole over $\Sigma\times R^+$ based on the Hermitian-Yang-Mills type structure of KW monopole equation. This is based on joint works with Rafe Mazzeo.
Elliott Wolf & Alex Woolf : CONVEX-OPTIMIZING THE POWER GRID
- Undergraduate Seminars ( 251 Views )The addition of renewable energy sources, whose power production cannot be scheduled, has created increasing gaps between instantaneous electricity supply and electricity demand. Sometimes the grid is oversupplied with energy, requiring zero-marginal-cost sources of power to be shut or energy to be bled off of the grid. Other times there is insufficient electricity, requiring high-marginal-cost sources of electricity to be switched on or consumers to curtail their demand. The current state of the grid has led various utilities and power consumers deploy capital-intensive energy storage, such as lithium-ion batteries, to better-match grid supply with grid demand. We present a method to add large-scale energy storage to the power grid using only sensors, software modifications to the control systems of large industrial refrigeration systems, and mathematical optimization. Our talk will address the required instrumentation, the physics necessary to understand applicable thermal constraints, and numerical methods used to determine a mathematically optimal-discharge schedule. We further discuss the economics of the US power grid, "war stories"of doing complex mathematics in a large industrial setting and the effects of various Federal Energy Regulatory Commission and California Public Utility Commission on our efforts.
Subhankar Dey : Cable knots are not thin
- Geometry and Topology ( 246 Views )Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.
Jun Kitagawa : A convergent Newton algorithm for semi-discrete optimal transport
- Applied Math and Analysis ( 246 Views )The optimal transport (Monge-Kantorovich) problem is a variational problem involving transportation of mass subject to minimizing some kind of energy, and it arises in connection with many parts of math, both pure and applied. In this talk, I will discuss a numerical algorithm to approximate solutions in the semi-discrete case. We propose a damped Newton algorithm which exploits the structure of the associated dual problem, and using geometric implications of the regularity theory of Monge-Amp{\`e}re equations, we are able to rigorously prove global linear convergence and local superlinear convergence of the algorithm. This talk is based on joint work with Quentin M{\’e}rigot and Boris Thibert.
Tye Lidman : Homology cobordisms with no 3-handles
- Geometry and Topology ( 243 Views )Homology cobordisms are a special type of manifold which are relevant to a variety of areas in geometric topology, including knot theory and triangulability. We study the behavior of a variety of invariants under a particular family of four-dimensional homology cobordisms which naturally arise from Stein manifolds. This is joint work with Ali Daemi, Jen Hom, Shea Vela-Vick, and Mike Wong.
Daniel Stern : Scalar curvature and circle-valued harmonic maps
- Geometry and Topology ( 242 Views )We introduce a new tool for relating the scalar curvature of a Riemannian manifold to its global geometry and topology, based on the study of level sets of harmonic functions and harmonic maps to the circle. We will explain how these ideas lead to simple new proofs and improvements upon some well-known results in three-manifold geometry and general relativity, previously studied primarily via minimal surface and Dirac operator methods.
Mainak Patel : The Essential Role of Phase Delayed Inhibition in Decoding Synchronized Oscillations within the Brain
- Undergraduate Seminars ( 242 Views )The widespread presence of synchronized neuronal oscillations within the brain suggests that a mechanism must exist that is capable of decoding such activity. Two realistic designs for such a decoder include: 1) a read-out neuron with a high spike threshold, or 2) a phase-delayed inhibition network motif. Despite requiring a more elaborate network architecture, phase-delayed inhibition has been observed in multiple systems, suggesting that it may provide inherent advantages over simply imposing a high spike threshold. We use a computational and mathematical approach to investigate the efficacy of the phase-delayed inhibition motif in detecting synchronized oscillations, showing that phase-delayed inhibition is capable of detecting synchrony far more robustly than a high spike threshold detector. Furthermore, we show that in a system with noisy encoders where stimuli are encoded through synchrony, phase-delayed inhibition enables the creation of a decoder that can respond both reliably and specifically to a stimulus, while a high spike threshold does not.